Varying The Gibbons-Hawking Term

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The Gibbons Hawking boundary term is given as $S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta$.
I want to calculate its variation with respect to the induced boundary metric, $h_{\mu \nu}$.

The answer (given in eqns 6&7 of http://arxiv.org/pdf/hep-th/9902121v5.pdf) is $\delta S_{GHY} = \frac{1}{16 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} (\Theta^{\mu \nu} - \Theta \gamma^{\mu \nu}) \delta \gamma_{\mu \nu}$

My attempt to obtain this goes as follows:

$$\delta (\sqrt{-\gamma} \Theta) = (\delta \sqrt{-\gamma}) \Theta + \sqrt{-\gamma} \delta \Theta$$
$$=\frac{1}{2} \sqrt{-\gamma} \gamma^{\mu \nu} \delta \gamma_{\mu \nu} \Theta+ \sqrt{-\gamma} \delta \Theta$$
$$= \frac{1}{2} \sqrt{-\gamma} \left( \Theta \gamma^{\mu \nu} + 2 \frac{\delta \Theta}{\delta \gamma_{\mu \nu}} \right) \delta \gamma_{\mu \nu}$$

I do not understand how to vary the extrinsic curvature, $\Theta = \gamma_{\mu \nu} \Theta^{\mu \nu} = \gamma_{\mu \nu} \nabla^\mu N^\nu$. Can anyone help me with this?

There is a post that explains how to vary the normal vector (http://physics.stackexchange.com/qu...ibbons-hawking-york-boundary-term/10628#10628) that is probably useful although it is varying with respect to the full spacetime metric and not the induced boundary metric but the end result looks the same so I imagine the technique is correct. I just do not understand how it works!

Thanks.

 

[nuclear_chris]

I would suggest expanding the summation over mu and nu and then applying the variation operator.